An insurance company sells a 1-year term life insurance policy to an 81-year-old woman. The woman pays a premium of 2700. If she dies within 1 year, the company will pay 92,000 to her beneficiary. According to the U.S. Centers for Disease Control and Prevention, the probability that an 81-year-old woman will be alive year later is . Let X be the profit made by the insurance company.

Question

Find the probability distribution.

Answer 1

To find the probability distribution for the profit \( X \) made by the insurance company from selling the life insurance policy, we need to consider two possible outcomes: the woman lives for the year, and the woman dies within the year.

1. Define the associated probabilities:

   • Let \( p \) be the probability that the 81-year-old woman will die within the year.
   • The probability that she will be alive after 1 year is then \( 1 - p \).

   According to the question, we don't have the exact value of \( p \) from the CDC. However, we can denote the probability of survival as \( p_{\text{alive}} \). Thus, \( p = 1 - p_{\text{alive}} \).

2. Profit outcomes:

   • If the woman lives for the year:

     • The insurance company keeps the premium of \( 2700 \) since no payout is made.
     • Profit \( X = 2700 \).

   • If the woman dies within the year:

     • The insurance company has to pay out \( 92,000 \).
     • Profit \( X = 2700 - 92000 = -89200 \).

3. Construct the probability distribution: Let’s define the outcomes and their associated probabilities:

• \( X = 2700 \) with probability \( p_{\text{alive}} \)
• \( X = -89200 \) with probability \( p_{\text{die}} = 1 - p_{\text{alive}} \)

4. Summary of the probability distribution: We can summarize this as follows:

\[ \begin{array}{|c|c|c|} \hline X & \text{Probability} \ \hline 2700 & p_{\text{alive}} \ -89200 & 1 - p_{\text{alive}} \ \hline \end{array} \]

Conclusion

The profit \( X \) has a discrete probability distribution characterized with \( 2700 \) with a probability of survival (denote as \( p_{\text{alive}} \)) and \(-89200\) with a probability of dying (denote as \( 1 - p_{\text{alive}} \)). To fully determine this distribution, you would substitute \( p_{\text{alive}} \) with the actual probability from the CDC's data for an 81-year-old woman's survival rate.

Answer 2

the probability is 0.9714
now compute

Answer 3

Given that the probability that an 81-year-old woman will be alive one year later is \( p_{\text{alive}} = 0.9714 \), we can calculate the probabilities for the profit outcomes.

1. Calculate the probability of dying: \[ p_{\text{die}} = 1 - p_{\text{alive}} = 1 - 0.9714 = 0.0286. \]

2. Profit outcomes:

   • If the woman lives (probability \( p_{\text{alive}} = 0.9714 \)):

     • Profit \( X = 2700 \)

   • If the woman dies (probability \( p_{\text{die}} = 0.0286 \)):

     • Profit \( X = -89200 \)

3. Summary of the probability distribution: \[ \begin{array}{|c|c|c|} \hline X & \text{Probability} \ \hline 2700 & 0.9714 \ -89200 & 0.0286 \ \hline \end{array} \]

Final Result

• Profit \( X \):
  • \( 2700 \) with probability \( 0.9714 \)
  • \( -89200 \) with probability \( 0.0286 \)

Expected Profit Calculation

To further examine the profitability, we can calculate the expected profit \( E(X) \):

\[ E(X) = (2700 \times 0.9714) + (-89200 \times 0.0286) \]

Calculating the values:

\[ E(X) = (2700 \times 0.9714) + (-89200 \times 0.0286) \] \[ = 2622.78 - 2545.92 \] \[ = 76.86 \]

Conclusion

The expected profit \( E(X) \) for the insurance company from this policy is approximately $76.86. The insurance company can expect to make a profit of roughly $76.86 on average for each policy sold to an 81-year-old woman with these parameters.

The probability distribution has been established, along with the expected value of the profit from the policy.

Answer 4

its not -89,200 its 89,300. Now compute the new expected value of the profit

Answer 5

Thank you for the correction regarding the profit related to the woman's death. Let's update the calculations accordingly, using the correct value of \( 89,300 \) for the payout.

Updated Profit Outcomes

1. If the woman lives (probability \( p_{\text{alive}} = 0.9714 \)):

   • Profit \( X = 2700 \)

2. If the woman dies (probability \( p_{\text{die}} = 0.0286 \)):

   • Profit \( X = 2700 - 89300 = -86600 \)

Summary of the Probability Distribution:

\[ \begin{array}{|c|c|c|} \hline X & \text{Probability} \ \hline 2700 & 0.9714 \ -86600 & 0.0286 \ \hline \end{array} \]

Expected Profit Calculation

To calculate the expected profit \( E(X) \):

\[ E(X) = (2700 \times 0.9714) + (-86600 \times 0.0286) \]

Calculating each term:

1. Live Outcome: \[ 2700 \times 0.9714 = 2622.78 \]

2. Die Outcome: \[ -86600 \times 0.0286 = -24751.80 \]

Now, sum these to find the expected profit:

\[ E(X) = 2622.78 - 24751.80 \] \[ E(X) = -22129.02 \]

Conclusion

The expected profit \( E(X) \) for the insurance company from this policy is approximately \(-22,129.02\). This indicates that, on average, the insurance company would incur a loss of about $22,129.02 for each policy sold under these conditions.

Answer 6

b) Find the expected value of the profit.

Answer 7

To find the expected value of the profit from the life insurance policy, let's outline the outcomes again considering the corrected payout.

Outcomes and Probabilities

1. If the woman lives (probability \( p_{\text{alive}} = 0.9714 \)):

   • Profit \( X = 2700 \)

2. If the woman dies (probability \( p_{\text{die}} = 0.0286 \)):

   • Profit \( X = 2700 - 89300 = -86600 \)

Summary of the Probability Distribution

\[ \begin{array}{|c|c|c|} \hline X & \text{Probability} \ \hline 2700 & 0.9714 \ -86600 & 0.0286 \ \hline \end{array} \]

Expected Value Calculation

To compute the expected value \( E(X) \): \[ E(X) = (2700 \times 0.9714) + (-86600 \times 0.0286) \]

Let's calculate each term:

1. Live Outcome: \[ 2700 \times 0.9714 = 2622.78 \]

2. Die Outcome: \[ -86600 \times 0.0286 = -24751.80 \]

Combining the Results

Now, we can combine these results to find the expected profit: \[ E(X) = 2622.78 - 24751.80 \] \[ E(X) = -22129.02 \]

Conclusion

The expected value of the profit \( E(X) \) from selling this life insurance policy is approximately \(-22,129.02\). This suggests that, on average, the insurance company would incur a loss of about $22,129.02 for each policy sold under these parameters.